MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ofmresval Structured version   Unicode version

Theorem ofmresval 6490
Description: Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
Hypotheses
Ref Expression
ofmresval.f  |-  ( ph  ->  F  e.  A )
ofmresval.g  |-  ( ph  ->  G  e.  B )
Assertion
Ref Expression
ofmresval  |-  ( ph  ->  ( F (  oF R  |`  ( A  X.  B ) ) G )  =  ( F  oF R G ) )

Proof of Theorem ofmresval
StepHypRef Expression
1 ofmresval.f . 2  |-  ( ph  ->  F  e.  A )
2 ofmresval.g . 2  |-  ( ph  ->  G  e.  B )
3 ovres 6379 . 2  |-  ( ( F  e.  A  /\  G  e.  B )  ->  ( F (  oF R  |`  ( A  X.  B ) ) G )  =  ( F  oF R G ) )
41, 2, 3syl2anc 659 1  |-  ( ph  ->  ( F (  oF R  |`  ( A  X.  B ) ) G )  =  ( F  oF R G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842    X. cxp 4940    |` cres 4944  (class class class)co 6234    oFcof 6475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-xp 4948  df-res 4954  df-iota 5489  df-fv 5533  df-ov 6237
This theorem is referenced by:  psradd  18247  dchrmul  23796  ldualvadd  32128
  Copyright terms: Public domain W3C validator